"That number of open flights continues to decrease thanks to our pilots who are stepping up to the plate and picking up trips to ensure customers are taken care of," said American Airlines' spokesman Matt Miller.

The airline offered pilots 150 percent of their hourly pay to sign up to work on the understaffed flights.

A representative of the union representing some 15,000 American Airlines' pilots told CNBC that he is trying to confirm the figures from the airline on the scheduling platform.

The airline said it doesn't expect to cancel any flights because of the problem.

If the situation remains unsolved, the pilot shortage could be costly for the airline.

The Association of Professional Flight Attendants, which represents about 26,000 American Airlines cabin crew members, said it would demand pay for the employees who signed up to work the schedules that lack pilots if their flights were canceled. Flight attendants only get paid when the flight begins.